Variously designed apparatuses of the above-described type are known from practice. With such an apparatus, one tries to wind a fiber strand onto a bobbin as rapidly as possible. However, increasing winding rates frequently require sacrificing the reliability of winding operations. Many of the known apparatuses thus leave much to be desired, in terms of both winding rates and the reliability of winding operations. Special problems arise when winding fiber strands that have been impregnated with a fusible thermoplastic, in which case, both the problem of reliably winding the ends of fiber strands onto bobbins and the problem of handling the ends of fiber strands remaining on bobbins following termination of winding operations and conduct of cutting operations arise, where the hardened ends of fiber strands will have to be remelted and subsequently ironed either onto bobbins or fiber strands that have been wound onto them, using elaborate ironing devices, is known. Cutting off and removing the ends of fiber strands is also known. However, cutting off waste and disposal thereof are also complex matters. Many disadvantages of the known apparatuses are also due to the belief that fiber strands must be continuously fed, without interruption, when winding them onto several bobbins in succession. The apparatuses known from actual practice are thus in need of improvement.